Let V be an orthogonal representation of a compact Lie group G and let S(V),D(V) be the unit sphere and disc of V, respectively. If F: V → ℝ is a G-invariant C¹-map then the G-equivariant gradient C⁰-map ∇F: V → V is said to be admissible provided that ${\left(\nabla F\right)}^{-1}\left(0\right)\cap S\left(V\right)=\varnothing$. We classify the homotopy classes of admissible G-equivariant gradient maps ∇F: (D(V),S(V)) → (V,V∖0).

Let G be a finite group, ${𝕆}_G$ the category of canonical orbits of G and $A:{𝕆}_G\to 𝔸$b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with $Ex{t}^{n-1}(A,A\otimes A)$. Then the case $G={\mathbb{Z}}_{p^k}$ leads to an example of infinitely...